Well it all depends on the outcome of the draft for me. Every year I am always incredibly interested and optimistic about the draft and start checking mocks before the season even ends. So the period before the draft seems to go by so slowly, but at least you can project who is gonna land where to kill time.
After the draft there truly is nothing interesting happening, signing rookies isn't exactly thriling to me. But if we have a good draft (such as two years ago) I can bask in the glory of that successful draft. Last year, when I didn't like our draft coupled with the lack of any interesting news was horrible.
So in summation, I guess I'd have to say the weeks after the draft are the worst time of the year for the NFL.